Verifying a Function as a Solution of Logistic Differential Equation

Question

\begin{cases} \frac {dP}{dt} = rP(t)(1-\frac {P(t)}{K}) ,t \geq 0 \\ P(0) = P_o \end{cases}

$r, K$ and $P_o$ are positive constants.

We say that $P(t), t \geq0$ is a solution of the above equation if it is differentiable at every $t \geq$0 (at $t=0$ it only needs to have one-sided derivative) and $P$ together with its derivative $\frac {dP}{dt}$ satisfy both lines of the equation.

Verify that the function:
$$P(t) = \frac {KP_0 e^r{^t}}{(K-P_0)+P_0e^r{^t}} , t \geq0$$ is the solution

My attemp:
\begin{align} \frac {dP}{dt}&= rP(t)(1-\frac {P(t)}{K})\\ rP(t)dt& = \frac {dP}{1-\frac {P(t)}{K}} \end{align} I am not sure whether I am on the right path of solving this problem and how to continue it. Can you help explain it to me in details? It would really help clear my confusion. Thank you.

Answer 1

To verify that $x(t)$ is a solution to $\frac{\text dx}{\text dt}(t) = f(t,x)$, simply calculate $\frac{\text dx}{\text dt}(t)$ from the given $x(t)$ and substitute both into the equation.

To verify that $x(t)$ satisfies the condition $x(t_0) = x_0$, substitute $t_0$ into $x(t)$ and substitute the result into the condition equation.

Answer 2

You may find the substitution $Q:=P^{-1}$ helps. You need to show $Q=1/K+(1/P_0-1/K)e^{-rt}\implies\dot{Q}=r/K-rQ$.